The total volume of bound charge is zero.
Explanation:
We have to the volume and surface bounded charge densities.
ρb = - Δ . p = - Δ .k ([tex]x^{X}[/tex] +[tex]y^{Y}[/tex] +[tex]x^{Y}[/tex])
= - 3k
On the top of the cube the surface charge density is
σb = p . z
= [tex]\frac{ka}{2}[/tex]
By symmetry this holds for all the other sides. The total bounded charge should be zero
Qtot = (-3k)a³ + 6 . [tex]\frac{ka}{2}[/tex] . a² = 0
σb = -3K σb = [tex]\frac{ka}{2}[/tex]
Qtot = 0
Hence, the total volume of bound charge is zero.
